Probabilistic Couplings For Probabilistic Reasoning

This thesis explores proofs by coupling from the perspective of formal verification. Long employed in probability theory and theoretical computer science, these proofs construct couplings between the output distributions of two probabilistic processes. Couplings can imply various probabilistic relational properties, guarantees that compare two runs of a probabilistic computation. To give a formal account of this clean proof technique, we first show that proofs in the program logic pRHL (probabilistic Relational Hoare Logic) describe couplings. We formalize couplings that establish various probabilistic properties, including distribution equivalence, convergence, and stochastic domination. Then we deepen the connection between couplings and pRHL by giving a proofs-as-programs interpretation: a coupling proof encodes a probabilistic product program, whose properties imply relational properties of the original two programs. We design the logic xpRHL (product pRHL) to build the product program, with extensions to model more advanced constructions including shift coupling and path coupling. We then develop an approximate version of probabilistic coupling, based on approximate liftings. It is known that the existence of an approximate lifting implies differential privacy, a relational notion of statistical privacy. We propose a corresponding proof technique---proof by approximate coupling---inspired by the logic apRHL, a version of pRHL for building approximate liftings. Drawing on ideas from existing privacy proofs, we extend apRHL with novel proof rules for constructing new approximate couplings. We give approximate coupling proofs of privacy for the Report-noisy-max and Sparse Vector mechanisms, well-known algorithms from the privacy literature with notoriously subtle privacy proofs, and produce the first formalized proof of privacy for these algorithms in apRHL. Finally, we enrich the theory of approximate couplings with several more sophisticated constructions: a principle for showing accuracy-dependent privacy, a generalization of the advanced composition theorem from differential privacy, and an optimal approximate coupling relating two subsets of samples. We also show equivalences between approximate couplings and other existing definitions. These ingredients support the first formalized proof of privacy for the Between Thresholds mechanism, an extension of the Sparse Vector mechanism

Apperceptive patterning: Artefaction, extensional beliefs and cognitive scaffolding

In “Psychopower and Ordinary Madness” my ambition, as it relates to Bernard Stiegler’s recent literature, was twofold: 1) critiquing Stiegler’s work on exosomatization and artefactual posthumanism—or, more specifically, nonhumanism—to problematize approaches to media archaeology that rely upon technical exteriorization; 2) challenging how Stiegler engages with Giuseppe Longo and Francis Bailly’s conception of negative entropy. These efforts were directed by a prevalent techno-cultural qualifier: the rise of Synthetic Intelligence (including neural nets, deep learning, predictive processing and Bayesian models of cognition). This paper continues this project but first directs a critical analytic lens at the Derridean practice of the ontologization of grammatization from which Stiegler emerges while also distinguishing how metalanguages operate in relation to object-oriented environmental interaction by way of inferentialism. Stalking continental (Kapp, Simondon, Leroi-Gourhan, etc.) and analytic traditions (e.g., Carnap, Chalmers, Clark, Sutton, Novaes, etc.), we move from artefacts to AI and Predictive Processing so as to link theories related to technicity with philosophy of mind. Simultaneously drawing forth Robert Brandom’s conceptualization of the roles that commitments play in retrospectively reconstructing the social experiences that lead to our endorsement(s) of norms, we compliment this account with Reza Negarestani’s deprivatized account of intelligence while analyzing the equipollent role between language and media (both digital and analog)

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs